I need x or y for the triangle area that forms between the vertical axis(y) and the function y=100+2x where the area is equal to 2500.

so I used for condition to the linear function:

knowing that the triangle area in this case should be like:
x*y/2=area, so:

x-100*y/2=2500
x-100*y=5000
y=5100/x

and then:

5100/x=100+2x
5100=100+2x*x
5000=2x^2
sqrt(2500)=x
50=x

the weird thing is that works for any area, and gives me the correct result for what I'm looking for, wich is x=50 and y=f(50)=200, if the area is calculated as is shown in the condition: 200-100*50/2=2500 !

When you multiply both sides of an equation by something, you have to distribute. Put parentheses around it and you'll see: $(5100/x)=(100+2x)$ so $5100=(100+2x)x=100x+2x^2$.
–
icurays1Nov 20 '12 at 18:23

2 Answers
2

In order to remove the $x$ from the denominator you correctly decided to multiply by $x$ on both sides. However, when you do this you should get $$x \frac{5100}{x} = x(100 + 2x)$$
And multiplying $x$ through the equation becomes
$$ 5100 = 100x + 2x^2$$
and now I'm sure you can solve it!

EDIT: To address the edit in your question,
be careful when setting up the area of this triangle because it is actually
$$\frac{1}{2} x(y-100)$$
since the bottom part of our triangle is located at $y = 100$.

And to solve this area to be 2500, we would plug in $$ \frac{1}{2} x(y-100) = 2500$$
And we know that $y = 100 + 2x$, so plugging that in for $y$ gives us $$ \frac{1}{2}x(100 + 2x - 100) = 2500$$
and canceling out the $100$s and multiplying through by the $x$ and $\frac{1}{2}$ gives $$ x^2 = 2500$$ or $x = 50$.

You start with the equation $\frac{5100}{x} = 100 + 2x.$ If the left side and the right side are equal, then I can do the same to both sides and they'll still be equal. Let's multiply both sides by $x$. I get $5100 = 100x + 2x^2.$ Bringing all of the terms over to one side, we get $2x^2 + 100x - 5100 = 0.$ Next, notice that there is a common factor: $2(x^2 + 50x - 2550) = 0.$ Finally, we use the quadratic formula where $a = 1,$ $b = 50$ and $c = -2550$. We have:

It seems that the website was correct. Notice that $-5(5\pm \sqrt{127}) = -25 \mp 5\sqrt{127}.$ If you don't see why $\sqrt{12700} = 10\sqrt{127}$ then notice that $12700 = 2^2 \times 5^2 \times 127$, where $127$ is prime and so: